Möbius Field Theory: Klein-bottle spacetime, Z6 holonomy, and a structured program relating topology to Standard Model gauge structure, couplings, mixing, and mass templates

We present Möbius Field Theory (MFT), a framework identifying four-dimensional spacetime with a compact non-orientable Klein bottle carrying Z₆ = Z₂ × Z₃ holonomy. From this geometric postulate together with one physically motivated energy scale, the framework derives the Standard Model gauge group, exactly three fermion generations, a self-consistency relation for α matching 1/137. 036, leading-order dark energy fraction Ω_Λ = 2/3 (2. 6% vs Planck 2018), Newton's constant via ℓPl² = (π+2) a³/LH (0. 09% accuracy), all four CKM parameters (1. 2–4. 5% errors), the electron mass, Bohr radius, and hydrogen ionization energy (0. 03% accuracy), and the neutrino mass scale from crosscap excitation m_ν = ε e^-8π ≈ 3. 45 meV, consistent with the PDG-input effective Majorana mass window 1. 48–3. 68 meV (normal ordering, m₁=0, Majorana phases free, one free parameter ε), new quaternion algebra, projector structure, M_ν^++ derivation, and PMNS mechanism identification. Draft V12 (April 2026) adds the following Pmath results, all with full in-manuscript proofs: The crosscap operators T₁, T₂, T₃ on ℂ⁴⊗ℂ³ satisfy strict quaternion relations, establishing an SU (2) -type flavor algebra. At the lepton vacuum θ=3π/4, generators T₁ and T₂ are broken while T₃ survives — a spontaneous breaking pattern fixed entirely by the Clifford and permutation identities with no free parameters. When both charged lepton and neutrino mass matrices are Hermitian circulants, they share a common DFT eigenbasis F₃, forcing UPMNS = 𝕀. This result is a Pmath obstruction: observable PMNS mixing requires structure that breaks the common circulant frame. The Pin⁻ crosscap projectors Π± = ½ (𝕀 ∓ iUcross) — where Ucross² = -𝕀 — are Hermitian idempotents satisfying Π±² = Π±, Π₊Π₋ = 0, Π₊+Π₋ = 𝕀. The naive ansatz ½ (𝕀+Ucross) fails idempotency because Ucross² = -𝕀 ≠ +𝕀. The physical neutrino mass matrix M_ν^++ = Π₊ (𝕀₄⊗Mgen) Π₊ expands as (v/2) 𝕀₄⊗MS + (v/4) γ₅⊗C with MS, C dimensionless and explicitly computed. The C matrix has eigenvalues 0, +3, -3 and is non-circulant, breaking the F₃ diagonalization and generating observable mixing with zero free parameters. The zero mode m₁=0 is preserved exactly under Π₊ projection — a fourth independent derivation of the massless lightest neutrino. Eleven Phase E dead ends (DE-V12-01 through DE-V12-11) are documented with stable labels, including the Type I Seesaw (incompatible with the direct crosscap mass mechanism by eleven orders of magnitude) and the naive real projector (fails idempotency). Epistemic status is explicit throughout. Pmath results are proven from stated definitions. CMFT results are conditional on the Connes-Lott product form. Hphys results are physically motivated but not derived from SMFT. Explicit PMNS mixing angles require V13 (diagonalization of M_ν^++ with charged lepton sector and Minkowski signature). The manuscript does not claim completeness — it claims honesty about what is proven, what is motivated, and what remains open. Carried forward from V10/V10. 1 and V11: two-loop holonomy split, Koide-Clifford mutual exclusion, S₃ fixed-point generation label, crosscap mass mechanism Scrosscap=8π giving m_ν = ε·e^-8π ≈ 3. 45 meV (consistent with PDG mₑe window 1. 48–3. 68 meV under normal ordering with m₁=0). V11 frozen at companion upload. V9 Zenodo 10. 5281/zenodo. 19244541. V10/V10. 1 Zenodo 10. 5281/zenodo. 19412329.